The Direct Construction of Lyapunov Functions for Nonlinear Systems

Abstract The direct construction of Lypunov functions is achieved using a new set construction method by which system trajectories are warped to form level sets for the functions. Applications to linear and nonlinear second order systems are demonstrated, including the stability analysis of a nonlinear guidance law for UAVs. Analysis of nonlinear systems with bounded inputs is also achieved by producing a level set that bounds the system state once a system trajectory enters the set. Thus the evaluation of bounded input-bounded output (BIBO) stability for nonlinear systems is realized. Finally, the method is generalized to higher order systems and three dimensional invariant sets are constructed for a third order nonlinear UAV guidance system.